If the line `vec(OR)` makes angles ` theta_(1),theta_(2),theta_(3)` with the planes ` XOY, YOZ, ZOX` respectively , then ` cos^(2)theta_(1)+cos^(2)theta_(2)+cos^(2)theta_(3)` is equal to

Explanation

Let `P(x, y, z)` be a point on the line and `OP = r(gt o)`
Draw seg PN perpendicular to the XOY-plane.
Since line ON lies in XOY-plane and the line OP makes an angle `theta_(1)` with XOY-plane
`therefore mangleNOP = theta_(1)` and N is `(x, y, 0)`.
`therefore ON = sqrt(x^(2)+y^(2))`and
`OP = r = sqrt(r^(2) + y^(2) +z^(2)).`
ON = projection of seg OP on the XOY-plane = r `cos theta_(1)`
`therefore cos theta_(1) = (ON)/(r ) = (sqrt(^(2) + y^(2)))/(r )`
Similarly, we can show that
`cos theta_(2)=(sqrt(y^(2)+z^(2)))/(r)andtheta_(3)=(sqrt(z^(2)+x^(2)))/(r)`
`therefore cos^(2)theta_(1)+cos^(2)theta_(2)+cos^(2)theta_(3)=(x^(2)+y^(2))/(r^(2))+(y^(3)+z^(2))/(r^(2))+(z^(2)+x^(2))/(r^(2))`
`=(2(x^(2)+y^(2)+z^(2)))/(r^(2))=(2r^(2))/(r^(2))=2.`